document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1061077>Question 1061077</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the domain of the rational function.\r<BR>\n" );
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document.write( "f(x) = 2x^2-4 / 3x^2+6x-45 \r<BR>\n" );
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document.write( "    a. {x|x &#8800; -3, x &#8800; 5}<BR>\n" );
document.write( "    b. {x|x &#8800; 3, x &#8800; -5}<BR>\n" );
document.write( "    c. all real numbers<BR>\n" );
document.write( "    d. {x|x &#8800; 3, x &#8800; -3, x &#8800; -5}<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #675902 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=675902\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The rational function should be written as<BR>\n" );
document.write( "f(x) = (2x^2-4) / (3x^2+6x-45) =<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%5E2-4%29%2F%283x%5E2%2B6x-45%29 ALIGN=MIDDLE  ALT=\"%282x%5E2-4%29%2F%283x%5E2%2B6x-45%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "with brackets to indicate that it is a quotient of two polynomials that must be evaluated before dividing,<BR>\n" );
document.write( "because I am pretty sure that<BR>\n" );
document.write( "f(x) = 2x^2-4 / 3x^2+6x-45 =<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2 ALIGN=MIDDLE  ALT=\"2x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4+%2F+3x%5E2 ALIGN=MIDDLE  ALT=\"-4+%2F+3x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%2B%22 ALIGN=MIDDLE  ALT=\"%22%2B%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x-45 ALIGN=MIDDLE  ALT=\"6x-45\"   DISABLED_style= \"max-width:90%; height:auto;\" > is not what was meant.<BR>\n" );
document.write( " <BR>\n" );
document.write( "TIPS:<BR>\n" );
document.write( "1) For multiple choice questions,<BR>\n" );
document.write( "when you do not need to \"show your work,\"<BR>\n" );
document.write( "you do not need to calculate much.<BR>\n" );
document.write( "2) Rational functions are defined everywhere except where the denominator is zero,<BR>\n" );
document.write( "and polynomials have an even number of real zeros equal or less than their degree.<BR>\n" );
document.write( "3) Polynomials are defined everywhere (meaning that their domain is all the real numbers),<BR>\n" );
document.write( "so if a polynomial is positive for a certain value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "and is negative for another value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "that polynomial is zero for some in-between value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "With the tips above, I first realize that the denominator, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%28x%29=3x%5E2%2B6x-45 ALIGN=MIDDLE  ALT=\"p%28x%29=3x%5E2%2B6x-45\"   DISABLED_style= \"max-width:90%; height:auto;\" > , is a polynomial of degree 2,<BR>\n" );
document.write( "so it may have two real zeros, or no real zeros.<BR>\n" );
document.write( "Then I realize that since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%28100%29=30645%3E0 ALIGN=MIDDLE  ALT=\"p%28100%29=30645%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%280%29=-45 ALIGN=MIDDLE  ALT=\"p%280%29=-45\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%28x%29 ALIGN=MIDDLE  ALT=\"p%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > must have one real zero between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=100 ALIGN=MIDDLE  ALT=\"x=100\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "but since real zeros come in pairs, it must have two real zeros.<BR>\n" );
document.write( "That eliminates answers c an d,<BR>\n" );
document.write( "so if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%28x%29 ALIGN=MIDDLE  ALT=\"p%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is zero for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3 ALIGN=MIDDLE  ALT=\"x=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the answer must be b,<BR>\n" );
document.write( "and otherwise the answer is a.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%283%29=3%2A3%5E2%2B6%2A3-45=3%2A9%2B18-45=27%2B18-45=45-45=0 ALIGN=MIDDLE  ALT=\"p%283%29=3%2A3%5E2%2B6%2A3-45=3%2A9%2B18-45=27%2B18-45=45-45=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so the answer is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28b%29 ALIGN=MIDDLE  ALT=\"highlight%28b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( " <BR>\n" );
document.write( "If it were not multiple choice answers to give you clues,<BR>\n" );
document.write( "you would have to find the zeros (if any) of the denominator.<BR>\n" );
document.write( "To do that you would try to solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2B6x-45=0 ALIGN=MIDDLE  ALT=\"3x%5E2%2B6x-45=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > by whatever method was easier for you.<BR>\n" );
document.write( "If you are good at factoring polynomials, you are in luck, because this one factors easily.<BR>\n" );
document.write( "In fact, you may have just written<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%282x%5E2-4%29%2F%283x%5E2%2B6x-45%29=%282x%5E2-4%29%2F%283%28x%5E2%2B2x-15%29%29=%282x%5E2-4%29%2F%283%28x%2B5%29%28x-3%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and that shows that the denominator is zero<BR>\n" );
document.write( "for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3 ALIGN=MIDDLE  ALT=\"x=3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-3=0 ALIGN=MIDDLE  ALT=\"x-3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-5 ALIGN=MIDDLE  ALT=\"x=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B5=0 ALIGN=MIDDLE  ALT=\"x%2B5=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That tells you <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-5 ALIGN=MIDDLE  ALT=\"x=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3 ALIGN=MIDDLE  ALT=\"x=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > are not part of the domain of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and that the answer is b.<BR>\n" );
document.write( "Otherwise, and in case where real zeros do not exist or are irrational,<BR>\n" );
document.write( "you may be tempted to solve the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%28x%29=0 ALIGN=MIDDLE  ALT=\"p%28x%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > using the quadratic formula.<BR>\n" );
document.write( "Solving <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2B6x-45=0 ALIGN=MIDDLE  ALT=\"3x%5E2%2B6x-45=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > using the quadratic formula is not fun.<BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2B6x-45=3%28x%5E2%2B2x-15%29 ALIGN=MIDDLE  ALT=\"3x%5E2%2B6x-45=3%28x%5E2%2B2x-15%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , solving <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B2x-15=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B2x-15=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a smarter idea<BR>\n" );
document.write( "that leads to less complicated calculations.<BR>\n" );
document.write( "Using the quadratic formula, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-2+%2B-+sqrt%282%5E2-4%2A1%2A%28-15%29%29%29%2F%282%2A1%29+=+%28-2+%2B-+sqrt%284%2B60%29%29%2F2+=+%28-2+%2B-+sqrt%2864%29%29%2F2+=+%28-2+%2B-+8%29%2F2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "would give you<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D=+%28-2+%2B8%29%2F2=6%2F2=3 ALIGN=MIDDLE  ALT=\"x%5B1%5D=+%28-2+%2B8%29%2F2=6%2F2=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D=+%28-2-8%29%2F2=%28-10%29%2F2=-5 ALIGN=MIDDLE  ALT=\"x%5B2%5D=+%28-2-8%29%2F2=%28-10%29%2F2=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );